Proof:
Consider a representation \(\phi \)  of \((Q,W)\)  with dimension vector \((n_1,n_2,n_3)\) ,
assumed to be stable for a King stability condition \(\theta =(\theta _1,\theta _2,\theta _3)\) .
Consider a cycle \(w\)  of the quiver: because \((Q,W)\)  gives a noncommutative crepant resolution of \({C}^3/{Z}_3\)  [1]}, the cycle \(w\)  is a central element of the Jacobian algebra of \((Q,W)\) , hence it defines an endomorphism of \(\phi \) . Because \(\phi \)  is stable, its only automorphisms are rescalings, hence \(w\)  acts as a scalar on \(\phi \) . Suppose now that \(\phi \)  is not of dimension vector \((n,n,n)\)  for \(n\in {N}^\ast \) : \(w\)  cannot be an automorphism (otherwise all its arrows would be isomorphisms, hence one would have \(n_1=n_2=n_3\) ), hence \(w\)  vanishes on \(\phi \) .
